Relational Exploration - Reconciling Plato and Aristotle

Relational Exploration – Reconciling Plato and Aristotle

Sebastian Rudolph Universitt Karlsruhe

Institut AIFB [email protected]

Abstract. We provide an interactive method for knowledge acquisition combin- ing approaches from description logic and formal concept analysis. Based on present data, hypothetical rules are formulated and checked against a description logic theory. We propose an abstract framework (Logical Domain Exploration) for this kind of exploration technique before presenting a concrete instantiation:

Relational Exploration. We give a completeness result and provide an overview about some application fields for our approach: machine learning, data mining, and ontology engineering.

1 Introduction

A plethora of research fields is concerned with the question of finding specifications for a given domain. Research areas like machine learning, frequent pattern discovery, and data mining in general aim at extracting these description on the basis of (examplary or complete) data sets – following the Aristotelian paradigm, that every conceptualization has to start from entities actually present. Other approaches intend to deduce these specifications from pre-specified theories – being somehow more Platonic by assuming the primacy of abstract ideas. The latter is the usual modus operandi e.g. in description logic or theorem proving.

We reconcile these two antagonistic approaches by combining techniques from two fields of knowledge representation: description logic (DL) and formal concept analysis (FCA).

In our work, we use DL formalisms for defining FCA attributes and FCA explo- ration techniques to obtain or refine DL knowledge representation specifications. More generally, DL exploits FCA techniques for interactive knowledge acquisition and FCA benefits from DL in terms of expressing relational knowledge.

In most cases, the process of conceptually specifying a domain cannot dispense of human contribution. However, although all information needed in order to describe a domain is in general implicitly present in an expert’s knowledge, the process of explicit formal specification may nevertheless be tedious and overstraining. Moreover, it might remain unclear whether a specification is complete, i.e., whether it covers all valid state- ments about the domain that can be expressed in the chosen specification language.

⋆This work has been supported by the European Commission under contract IST-2006-027595 NeOn, and by the Deutsche Forschungsgemeinschaft (DFG) under the ReaSem project.

Hence, we provide a method – called Relational Exploration (RE) – that organizes and structures the specification process by successively asking single questions to the domain expert in a way which minimizes the expert’s effort (in particular, it does not ask redundant questions) and guarantees that the resulting specification will be com- plete in the sense stated above. To present our work, which generalize the results from [1] and [2], we will proceed as follows: Section 2 provides a general framework for this kind of procedure, called Logical Domain Exploration. In Section 3, we shortly sketch the FCA basics necessary for our work and give an overview about attribute ex- ploration. Section 4 introduces the notions from description logics needed in this work.

In Section 5, we establish the correspondence between DL models and certain formal contexts, which enables us to apply FCA to DL. In Section 6, the RE algorithm is de- scribed in detail. Section 7 shows certain completeness properties of the knowledge acquired via RE. Section 9 displays direction for further work. In Section 8, we discuss our results and consider in which fields the presented technique could be applied.

2 The Epistemic Framework:

Logical Domain Exploration

Before engaging into the technical details, we sketch the overall setting for our ap- proach, which helps conveying the underlying idea and identifying the contributing components. Doing this on an abstract level, we also give an opportunity to relate alter- native approaches. This framework will be called Logical Domain Exploration.

Let ∆ be the considered domain of interest the elements of which we will call (DOMAIN)INDIVIDUALS. LetLbe a language the elements of which are calledFOR-

MULAE. We write∆|=ϕin order to state for a formulaϕthat it is valid in the domain.

Moreover let the setting be well-behaved in the way that whenever∆|=ϕis not true, there is a finite individual setΓ ⊆∆witnessing this (we then writeΓ †ϕand sayΓ

SPOILSϕ).

– The EXPERTis supposed to be “omniscient” wrt. the described domain and thus able to answer any question about it. In particular, he knows for all ϕ ∈ Land Γ ⊆∆whether∆|=ϕand whetherΓ†ϕ. Mostly, a human or a group of humans will take the role of the expert.

– The TERMINOLOGYconsists of a theoryTh ⊆ L^T about the domain consisting of axioms in some languageL^T ⊇ Land a reasoning functionality, i.e. for any statementϕ∈ L^T it can be decided whetherThentailsϕ.¹

– TheDATAconsists of a set of known or recorded individualsD ⊆∆and is endowed with a special querying capability, i.e., a procedure providing for anyϕ∈ La set Γ ⊆ DwithΓ †ϕif there exists one.

– The SCHEDULERcan be conceived as an automated procedure initiating and co- ordinating the ”information flow”. It links the other system components by asking questions, processing answers, and assuring that in the end all knowledge is ac- quired to quickly decide for anyϕ∈ Lwhether∆|=ϕ.

1Hereby, entailment is as usual defined in a model-theoretic way:This said to entailϕif any domain∆^′wherein all formulae ofThare valid also satisfies∆^′|=ϕ.

The system will operate as follows: We start with a (correct but in general incom- plete) terminological theoryTh ⊂ {ψ∈ L^T |∆|=ψ}and dataD ⊆∆. The scheduler now comes up with hypothetical formulae. Every such hypothetical formulaϕ∈ Lis passed both to the terminology and the data. The reasoning service of the terminology component checks whetherϕis entailed byTh. The data is queried for a spoiler ofϕ.

Since – due to the starting conditions – the theory is consistent with the data, we get three disjoint possible results:

– ϕis entailed byTh. In this case,ϕis valid in∆, which will be responded to the scheduler.

– Γ ∈ Dspoilsϕ. Then,ϕis not valid in∆and the scheduler will be provided with this negative answer.

– Neither of the previous cases occurs. Then, the current specification leaves room for either possibility and the domain expert will have to be asked this aboutϕ’s validity in question. If he confirms the validity ofϕin∆, it will be added toTh. If he denies it, he has to provide a spoilerΓ forϕ, which is then added to the data.

Note that querying the data and questioning the terminology can be done in either order or even in parallel. After finishing the procedure every formulaϕ ∈ Lwill either be a consequence of the resulting (updated) terminology or can be excluded via a spoiler present in the data (updated) data. The distinction betweenL(theEXPLORATION LAN-

GUAGE) andL^T (theTERMINOLOGICAL LANGUAGE) is motivated by the assumption that in most cases not all terminologically expressible axioms will be of interest but only those of a certain shape.

In the next chapters, we come down to an instance for the previously described framework for logical domain exploration: Relational Exploration.

3 Formal Concept Analysis

In our instantiation, the scheduler’s task will be carried out by an extension of the at- tribute exploration algorithm well established in FCA. This necessitates to briefly intro- duce some basic FCA notions. We mainly follow the notation introduced in [3] being the reference for FCA theory.

The basic notion FCA is built on is that of a formal context. It is a common claim in FCA that any kind of grounded data can be represented in this way.

Definition 1. A FORMAL CONTEXTKis a triple (G, M, I)with an arbitrary setG (calledOBJECTS), an arbitrary setM(calledATTRIBUTES), and a relationI⊆G×M (calledINCIDENCE RELATION). We readgImas: “objectghas attributem.” Further- more, letg^I :={m|gIm}.

The central means of expressing knowledge in FCA is via implications. Thus, in terms of the general framework from Section 2 the underlying language consists of implications on a fixed attribute set of atomic propositions.

Definition 2. LetM be an arbitrary set. AnIMPLICATIONonMis a pair(A, B)with A, B⊆M. To support intuition, we writeABinstead of(A, B).ABHOLDSin

a formal contextK= (G, M, I), if for allg∈Gwe have thatA⊆g^I impliesB⊆g^I. We then writeK|=AB.

ForC ⊆M and a setIof implications onM, letC^Idenote the smallest set with C⊆C^Ithat additionally fulfills

A⊆C^I implies B⊆C^I

for every implicationABinI.²IfC=C^I, we callCI-CLOSED. We sayI_ENTAILS ABifB⊆A^I.³

An implication setIwill be calledNON-REDUNDANT, if for any(AB)∈Iwe have thatB⊆A^I\{AB}.

An implication setIof a contextKwill be called COMPLETE, if every implication ABholding inKis entailed byI.

Iwill be called anIMPLICATION BASEof a formal contextKif it is non-redundant and complete.

Note that implication entailment is decidable in linear time ([4]). Therefore, know- ing a domain’s implication base allows fast handling of its whole implicational theory.

Moreover, for every formal context, there exists a canonical implication base ([5]).

The attribute exploration algorithm our work is based on was introduced in [6]. Due to space reasons, we omit to display it in detail and refer the reader to the literature.

Essentially, the following happens: the domain to explore is formalized as a formal contextK= (U, M, I). Usually, it is not known completely in advance. However, pos- sibly, some entities of the universeg∈U are already known, as well as their associated attributesg^I.

The algorithm now starts presenting questions of the form

“Does the implicationABhold in the contextK= (U, M, I)?”

to the human expert. The expert might confirm this. In this case,AB is archived as part ofK’s implicational baseIB. The other case would be thatABdoes not hold in(U, M, I). But then, there must exist ag ∈ U withA ∈g^I andB ∈ g^I. The expert is asked to input thisgandg^I.⁴The procedure terminates when the implicational knowledge of the universe is completely acquired, i.e., the implications of the formal context built from the entered counterexamples coincide with those entailed byIB.

In the approach presented here, we will exploit the capability of attribute exploration to efficiently determine an implicational theory. Notwithstanding, we extend the under- lying language⁵from purely propositional to certain DL expressions being introduced in the next section.

2Note, that this is well-defined, since the mentioned properties are closed wrt. intersection.

3Actually, this is a syntactic shortcut. Yet, it can be easily seen that this coincides with the usual entailment notion.

4Referring to the general framework we mention that in this special case the spoiler (called counterexample) is always a singleton set:{g} †AB.

5There exist already other language extensions, e.g. to Horn-logic with a bounded variable set, see [7].

4 Description Logic

We recall basic notions from DL, following (and recommending for further reading) [8].

Unlike the way DL is normally conceived, we use DL expressions to describe or specify one particular, fixed domain.

Thus, we will start our considerations by formally defining the kind of relational structure that we want to “talk about.”

Definition 3. AnINTERPRETATIONfor a setAof (PRIMITIVE)CLASS NAMESand a setRof ROLE NAMESis a pairI = (∆^I,(.)^I)where∆^I is some set and(.)^I is a function mapping class names to subsets of∆^Iand role names to subsets of∆^I×∆^I. Verbally, for some primitive class nameA,A^Iprovides all members of that class and for some role nameR,R^Iyields all ordered pairs “connected” by that role.

The DL languages introduced here provide constructors for defining new concept descriptions out of the primitive ones. Table 1 shows those constructors, their interpre- tation (as usual defined recursively), and their availabilities in the description logics considered here.

name interpretation FL0 EL FLE ALE

A atomic concept A^I × × × ×

⊤ universal concept ∆^I × × × ×

⊥ bottom concept ∅ × × × ×

¬A atomic negation ∆^I\A^I ×

C⊓Dconjunction C^I∩D^I × × × ×

∀R.C value restriction {δ| ∀ǫ: (δ, ǫ)∈R^I→ǫ∈C^I} × × ×

∃R.C existential quantification{δ| ∃ǫ: (δ, ǫ)∈R^I∧ǫ∈C^I} × × × Table 1. syntax and semantics of the DLs considered in this paper

In the sequel, we will in general speak of a description logicDLif the presented result or definition refers to anyDL ∈ {F L0,EL,F LE,ALE}.

Definition 4. LetIbe an interpretation andC,DbeDLconcept descriptions. We say CIS SUBSUMED BYDinI(written:C⊑I D) ifC^I ⊆D^I. This kind of subsumption statements is also calledGENERAL CONCEPT INCLUSION AXIOM(GCI). Moreover, we sayCandDareEQUIVALENTinI(written:C≡ID) ifC^I= D^I.

5 Subsumptions as Implications

Combinations of FCA and DL have already been described in several publications, e.g.

in [9], [10], and [11]. Our approach is motivated by [9] insofar as we use the same way of transferring a DL setting into a formal context by considering the domain individuals as objects and DL concept expressions as attributes.

Definition 5. Given an interpretationI = (∆^I,(.)^I) and a setM ofDL concept descriptions, we define the correspondingDL-CONTEXT

K^I(M) := (∆^I, M, I)

whereδIC:⇐⇒δ∈C^I, for allδ∈∆^IandC∈M.

The observation in the next theorem – though easy to see – is crucial for applying attribute exploration for the intended purpose.

Theorem 1. LetI be an arbitrary interpretation andK^I(M)a correspondingDL- context. Then for finiteC,D ⊆M, the implication

CD holds inK^I(M)if and only if⁶

C ⊑I

D.

In the sequel, we will exploit this correspondence in the following way: employ- ing the FCA exploration method allows us to collect all information that is valid in a (not explicitly given) interpretation and can be expressed byDL subsumptions with restricted maximal role depth⁷.

6 The Relational Exploration Algorithm

The algorithm we present here is an iterative one. In each step the maximal role depth of the consideredDLconcept descriptions will be incremented by one. In each step, the results from previous steps will be exploited in several ways.

In the worst case, the time needed for the attribute exploration algorithm is expo- nential with respect to the number of attributes. Thus, it is essential to see how the set of attributes can be reduced without losing completeness.

The first exploration step is aimed at clarifying the implicational interdependencies ofDLconcept descriptions with quantifier depth 0. Therefore, no roles occur yet and we start with

M0:=

{⊥} ∪ {A,¬A|A∈ A} ifDL=ALE {⊥} ∪ A otherwise.

In the actual exploration step – the interview-like procedure described in Section 3 – takes place with respect to the contextK^Ii =K^Ii(Mi). Every hypothetical implication ABforA,B ⊆ Mi presented to the expert has to be interpreted as question about the validity of

A ⊑I

B, and will be passed to the “answering components” as described in Section 2.

6We use

{C¹, . . . ,Cn}to abbreviateC1⊓. . .⊓Cⁿ. Moreover, let

{C}:= Cand {∅}:=

⊤.

7As usual, a concept description’s role depth indicates how deep quantifiers are nested in it.

The exploration step ends up with an implication baseIB_i, which – as we will prove in Section 7 – represents the complete subsumptional knowledge of the consid- ered domain up to role depthi.

For the next exploration step – incrementing the considered role depth – we have to stipulate the next attribute setMi+1. In case of the concept descriptions preceded by an existential quantification, the previously acquired implication baseIB_ican be used to reduce the number of attributes to consider, keeping the completeness property.

Mi+1:=M0

∪{∀R.C|R∈ R,C∈Mi}

∪{∃R.

C |R∈ R,C=C^IBi,⊥ ∈ C}

If consideringELorF L0, simply discard the second resp. third line from the defi- nition. In addition to minimizing the cardinality ofMi+1, we can accelerate the explo- ration process by providing implications onMi+1that are already known to be valid.

These are the following:

– {⊥}Mi+1,

– {(A)i+1 |A ∈ A}{(B)i+1 | B∈ B}for every implicationABfromIB_i (i.e., translate⁸all known implications fromMiintoMi+1),

– {∀R.A|A∈ A}{∀R.B|B∈ B}for every implicationABfromIB_i, – {∃R.

A}{∃R.

B}for allIB_i-closed setsA,B ⊆ MiwithA Bwhere there is noIB_i-closed setCwithACB, and

– {∃R.

A,∀R.A}{∃R.

(A ∪ {A})^IBi}for everyIB_i-closed setA ⊆Mi\ {A}and every concept descriptionA∈Mi.

With this attribute setMi+1 and the a-priori implications we start the next explo- ration step.

In theory, this procedure can be continued to arbitrary role depths. In some but not in all cases a complete acquisition of knowledge can be achieved. Yet in practice, with increasing role depth, the questions brought up by the exploration procedure will be increasingly numerous as well as less intuitional and thus difficult to answer for a human expert. So in many cases, one will restrict to small role depths.

7 Verification of the Algorithm

LetDLidenote the set of allDLconcept descriptions with maximal role depthi. Now we show a way how the validity of any subsumption onDLi can be checked by us- ing just the attribute setsM0, . . . , Mi as well as the corresponding implication bases IB₀, . . . ,IB_ion those sets. First, we will define functions that provide for any concept descriptionC∈ DLia set of attributesC ⊆ Misuch thatC ≡I

C. The following definitions and proofs are carried out forALE but can be easily adapted to the other DLs by simply removing the irrelevant parts.

8We will formally define and justify this translation(.)i+1in Section 7.

Definition 6. LetI be an interpretation and the corresponding sequences(Mi),(K^Ii) defined as above. Given the according sequenceIB₀, . . . ,IB_n of implication bases, we define a sequence of functionsτi :DLi→ P(DLi)in a recursive way:

τi(C) ={C} forC∈M0

τi(

C) =

{τi(C)|C∈ C}

τi(∀R.C) ={∀R.D|D∈τi−1(C)}

τi(∃R.C) =

{⊥} if⊥ ∈(τi−1(C))^IBi−1, {∃R.

(τi−1(C))^IBi−1} otherwise.

Moreover, let¯τi(C) := (τi(C))^IBifor allC∈ DLi. Note that by this definition, we also haveτ¯i(⊤) = ¯τi(

∅) =∅^IBi. Next, we have to show that the functions just defined behave in the desired way. The following lemma ensures thatτ¯iandτiindeed map toMi.

Lemma 1. SupposeC∈ DLi. Then we haveτi(C)⊆Miandτ¯i(C)⊆Mi.

Proof. Obviously,τ¯i(C)⊆Miwheneverτi(C)⊆Mi. We show the latter by induction on the role depth considering four cases:

– C∈ {A,¬A|A∈ A} ∪ {⊥}. Then by definitionC∈Mi.

– C =∃R.D. If⊥ ∈τ¯i−1(D), we getτi(C) =τi(∃R.D) ={⊥} ⊆Mi.

Now suppose⊥ ∈τ¯ⁱ⁻1(D). As immediate consequence of the induction hypothesis we haveτ¯i−1(D) ⊆ Mi−1. Sinceτ¯i−1 gives anIB_i−1-closed set, we have also

∃R.

¯

τi−1(D) ∈Mi, as a look to the constructive definition ofMiimmediately shows. Therefore,τi(C) =τi(∃R.D) ={∃R.

¯

τi−1(D)} ⊆Mi

– C =∀R.D. Again, our induction hypothesis yieldsτi−1(D)⊆Mi−1which implies {∀R.E | E ∈ τi−1(D)} ⊆ Mi due to the definition of Mi and therefore also τi(C) =τi(∀R.D) ={∀R.E|E∈τi−1(D)} ⊆Mi.

– C =

C. W.l.o.g., we presuppose that there is no conjunction outside the quantifier range in anyD ∈ C. So we haveτi(D) ⊆ Midue to the three cases above, and subsequently alsoτi(C) =τi(

C) = {τi(D)|C∈ C}

⊆Mi. 2

The next lemma and theorem show that in our fixed interpretationI, for any concept descriptionC∈ DLi, the entity sets fulfillingCon the one hand andτ¯i(C)as well as τi(C)on the other hand coincide.

Lemma 2. For anyC ⊆Mi, we have

C ≡I C^IBi. Proof. First, observe(

C)^I =

{(C)^I |C∈ C}=

{C^Ii |C∈ C}={δ∈∆^I | δ ∈ C^Ifor allC ∈ C}. Now, considerK^Ii. SinceIB_i is an implication base ofK^Ii, CC^IBi is an implication valid inK^Ii, ergo all objects ofK^Ii (being the individuals δ ∈ ∆^I) fulfill C ⊆ δ^Ii ⇒ C^IBi ⊆ δ^Ii. Therefore, one δ has all attributes from C exactly if it has all attributes from C^IBi. Finally, we have then {δ ∈ ∆^I | δ ∈ C^Ifor allC∈ C^IBi}=

{C^I|C∈ C^IBi}= (

C^IBi)^I.

2

Theorem 2. LetC∈ DLi. ThenC≡I

τi(C)≡I

¯ τi(C).

Proof. The second equivalence is a direct consequence of Lemma 2. We show the first one again via induction on the role depth:

– C∈ {A,¬A|A∈ A} ∪ {⊥}. Then, we trivially haveC^I = ( {C})^I. – C = ∃R.D. By induction hypothesis, we get D^I = (

¯

τi−1(D))^I, therefore (∃R.D)^I = (∃R.

¯

τi−1(D))^Iwhich by definition equals(

τi(∃R.D))˜ ^I. – C = ∀R.D. Again, by induction hypothesis, we get D^I = (

¯

τi−1(D))^I = {E^I | E∈τ¯i−1(D)}. Now, observe that the statement(δ,δ)˜ ∈R^I →δ˜∈D^I is equivalent to

E∈τi−1(D)

(δ,δ)˜ ∈ R^I → δ˜ ∈ E^I

and thus (∀R.D)^I = {δ | (δ,˜δ) ∈ R^I → δ˜ ∈ {D^I}} = {δ |

E∈τi−1(D)δ ∈ (∀R.E)^I} = {(∀R.E)^I | E ∈ τⁱ⁻1(D)} = (

{∀R.E | E ∈ τⁱ⁻1(D)})^I which by defini- tion is just(

τi(∀R.D))^I. – C =

C. Again, we can presume no conjunction outside the quantifier range in any D ∈ C. Then(

C)^I =

{(D)^I | D ∈ C} = {(

τi(D))^I | D ∈ C}

because of the cases shown before. Now, this is obviously the same as {(E)^I | E∈τi(D),D∈ C}= (

({τi(D)|D∈ C}))^I= (τi(

C))^I. 2

Using these propositions, we can easily provide a method to check – using only the closure operatorsIB₀, . . . ,IB_i– the validity of any subsumption onDLiwith respect to a fixed (but not explicitly known) interpretationI. It suffices to applyτ¯ion both sides and then check for inclusion.

Corollary 1. LetC1,C2∈ DLi. ThenC1⊑^I C2if and only ifτ¯i(C2)⊆¯τi(C1).

Proof. Due to Theorem 2,C1⊑IC2is equivalent to

¯

τi(C1)⊑I

¯

τi(C2). Accord- ing to Lemma 1, we haveτ¯i(C1)⊆Miandτ¯i(C2)⊆Mi. In view of Theorem 1, this means the same as the validity of the implicationτ¯i(C1)τ¯i(C2)inKi. Now, since the application of¯τalways gives a closed set with respect to all implications valid in Ki, this is equivalent toτ¯i(C2)⊆τ¯i(C1). 2 Finally, consider the function τi from Definition 6. It is easy to see that for any C∈Mi−1by calculatingτi(C)we get a singleton set{D}withD∈Mi. We then have evenC≡I D. For the sake of readability we will just writeD = (C)i. Roughly spoken, Dis just the “equivalentMi-version” ofC. Note that evaluatingτi does not need the implication baseIB_ibut onlyIB₀, . . . ,IBi−1. So we have provided the translation function we promised in Section 6.

8 Conclusion

We have introduced an interactive knowledge acquisition technique for finding DL-style subsumption statements valid in a domain of interest. Its outstanding properties are

– minimal workload for the domain expert (i.e., no redundant questions will be posed) and

– completeness of the resulting specification (any statement from the exploration lan- guage is known to hold or not to hold).

Several current fields of AI can benefit from the results presented here.

Ontology engineering would be the first to mention. Since based on DL formalisms, our method can obviously contribute to the development and refinement of ontologies.

RE can be used for an organized search for new GCIs⁹of a certain shape (namely those expressible byDLconcept descriptions). Clearly, the description logics nowaday’s on- tology specifications are based on are much more complex than any ofDL. Nonethe- less, our algorithm is still applicable since all of them incorporate the DLs considered as exploration language candidates. Hence, any of the existent reasoning algorithms for deciding subsumption (as for instance KAON2 [12] or FaCT [13], both capable of rea- soning inSHIQ(D)– see [14]) can be used for the terminology part. All information beyondDLwould then be treated as background knowledge and “hidden” from the exploration algorithm. As already pointed out, one major advantage of applying this technique is the guarantee that all valid axioms expressible as subsumption statements onDLwith a certain role depth will certainly be found and specified.

Another topic RE can contribute to is machine learning. The supervised case cor- responds almost directly to the RE algorithm – mostly one would have large data sets and (almost) empty theories in this setting. Yet also unsupervised machine learning can be carried out – by “short-circuiting” the expert such that every potential statement di- rected to him would be automatically confirmed. Essentially, the same would be the case for data mining tasks.

Finally, we are confident that an implementation of the RE algorithm will be a very helpful and versatile tool for eliciting information from various knowledge resources.

9 Future Work

So, as the very next step, we plan an implementation of the presented algorithm includ- ing interfaces for database querying as well as for DL reasoning. Applying this tool in the ontology engineering area will in turn enable us to investigate central questions concerning practical usability; in particular performance on real-life problems and scal- ability (being of unprecedented relevance in the Semantic Web Technologies sector), as well as issues concerning user acceptance will be of special interest for evaluation.

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